Question

How do you use the Intermediate Value Theorem and synthetic division to determine whether or not the following polynomial `P(x) = x^4 + 2x^3 + 2x^2 - 5x + 3` have a real zero between the numbers 0 and 1?

Answer 1

You can't.

Explanation:

You can use synthetic division (and the Remainder Theorem) to find

`P(0) = 3` (although using division seems like the long way to do this)

and `P(1) = 3`

If we one of these positive and the other negative, then, since `P(x)` is continuous, we could use the Intermediate Value Theorem to conclude that there is a zero in `(0,1)`.

The Intermediate Value Theorem can never be used to conclude that there is not a zero in an interval.

We could conclude that `P(x)` does not have exactly one zero between `0` and `1`. (It must have an even number of zeros in in `(0,1)`.)

To help explain why, here is the graph of a different polynomial function:

graph{ x^4 + 2x^3 + 2x^2 - 5x +1 [-3.067, 3.09, -1.17, 1.91]}

Call this polynomial function `Q(x)`

Notice that `Q(0) = 1` and also `Q(1) = 1` but his function does have a zero in `(0,1)`. (It has two of them.)